# §19.24 Inequalities

## §19.24(i) Complete Integrals

The condition $y\leq z$ for (19.24.1) and (19.24.2) serves only to identify $y$ as the smaller of the two nonzero variables of a symmetric function; it does not restrict validity.

 19.24.1 $\ln 4\leq\sqrt{z}R_{F}\left(0,y,z\right)+\ln\sqrt{y/z}\leq\tfrac{1}{2}\pi,$ $0, ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of first kind, $\pi$: the ratio of the circumference of a circle to its diameter and $\ln\NVar{z}$: principal branch of logarithm function Referenced by: §19.24(i) Permalink: http://dlmf.nist.gov/19.24.E1 Encodings: TeX, pMML, png See also: Annotations for §19.24(i), §19.24 and Ch.19
 19.24.2 $\tfrac{1}{2}\leq z^{-1/2}R_{G}\left(0,y,z\right)\leq\tfrac{1}{4}\pi,$ $0\leq y\leq z$, ⓘ Symbols: $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of second kind and $\pi$: the ratio of the circumference of a circle to its diameter Referenced by: §19.24(i) Permalink: http://dlmf.nist.gov/19.24.E2 Encodings: TeX, pMML, png See also: Annotations for §19.24(i), §19.24 and Ch.19
 19.24.3 $\left(\frac{y^{3/2}+z^{3/2}}{2}\right)^{2/3}\leq\frac{4}{\pi}R_{G}\left(0,y^{2% },z^{2}\right)\leq\left(\frac{y^{2}+z^{2}}{2}\right)^{1/2},$ $y>0$, $z>0$. ⓘ Symbols: $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of second kind and $\pi$: the ratio of the circumference of a circle to its diameter Referenced by: §19.24(i) Permalink: http://dlmf.nist.gov/19.24.E3 Encodings: TeX, pMML, png See also: Annotations for §19.24(i), §19.24 and Ch.19

If $y$, $z$, and $p$ are positive, then

 19.24.4 $\frac{2}{\sqrt{p}}(2yz+yp+zp)^{-1/2}\leq\frac{4}{3\pi}R_{J}\left(0,y,z,p\right% )\leq(yzp^{2})^{-3/8}.$ ⓘ Symbols: $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$: symmetric elliptic integral of third kind and $\pi$: the ratio of the circumference of a circle to its diameter Referenced by: §19.24(i) Permalink: http://dlmf.nist.gov/19.24.E4 Encodings: TeX, pMML, png See also: Annotations for §19.24(i), §19.24 and Ch.19

Inequalities for $R_{D}\left(0,y,z\right)$ are included as the case $p=z$.

A series of successively sharper inequalities is obtained from the AGM process (§19.8(i)) with $a_{0}\geq g_{0}>0$:

 19.24.5 $\frac{1}{a_{n}}\leq\frac{2}{\pi}R_{F}\left(0,a_{0}^{2},g_{0}^{2}\right)\leq% \frac{1}{g_{n}},$ $n=0,1,2,\dots$,

where

 19.24.6 $\displaystyle a_{n+1}$ $\displaystyle=(a_{n}+g_{n})/2,$ $\displaystyle g_{n+1}$ $\displaystyle=\sqrt{a_{n}g_{n}}.$ ⓘ Symbols: $n$: nonnegative integer, $a_{n}$: iterate and $g_{n}$: iterate Referenced by: §19.24(i) Permalink: http://dlmf.nist.gov/19.24.E6 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §19.24(i), §19.24 and Ch.19

Other inequalities can be obtained by applying Carlson (1966, Theorems 2 and 3) to (19.16.20)–(19.16.23). Approximations and one-sided inequalities for $R_{G}\left(0,y,z\right)$ follow from those given in §19.9(i) for the length $L(a,b)$ of an ellipse with semiaxes $a$ and $b$, since

 19.24.7 $L(a,b)=8R_{G}\left(0,a^{2},b^{2}\right).$ ⓘ Symbols: $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of second kind and $L(a,b)$: length Permalink: http://dlmf.nist.gov/19.24.E7 Encodings: TeX, pMML, png See also: Annotations for §19.24(i), §19.24 and Ch.19

For $x>0$, $y>0$, and $x\neq y$, the complete cases of $R_{F}$ and $R_{G}$ satisfy

 19.24.8 $\displaystyle R_{F}\left(x,y,0\right)R_{G}\left(x,y,0\right)$ $\displaystyle>\tfrac{1}{8}\pi^{2},$ $\displaystyle R_{F}\left(x,y,0\right)+2R_{G}\left(x,y,0\right)$ $\displaystyle>\pi.$

Also, with the notation of (19.24.6),

 19.24.9 $\frac{1}{2}\,g_{1}^{2}\leq\frac{R_{G}\left(a_{0}^{2},g_{0}^{2},0\right)}{R_{F}% \left(a_{0}^{2},g_{0}^{2},0\right)}\leq\frac{1}{2}\,a_{1}^{2},$

with equality iff $a_{0}=g_{0}$.

## §19.24(ii) Incomplete Integrals

Inequalities for $R_{-a}\left(\mathbf{b};\mathbf{z}\right)$ in Carlson (1966, Theorems 2 and 3) can be applied to (19.16.14)–(19.16.17). All variables are positive, and equality occurs iff all variables are equal.

### Examples

 19.24.10 $\frac{3}{\sqrt{x}+\sqrt{y}+\sqrt{z}}\leq R_{F}\left(x,y,z\right)\leq\frac{1}{(% xyz)^{1/6}},$ ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of first kind Referenced by: §19.9(ii) Permalink: http://dlmf.nist.gov/19.24.E10 Encodings: TeX, pMML, png See also: Annotations for §19.24(ii), §19.24(ii), §19.24 and Ch.19
 19.24.11 $\left(\frac{5}{\sqrt{x}+\sqrt{y}+\sqrt{z}+2\sqrt{p}}\right)^{3}\leq R_{J}\left% (x,y,z,p\right)\leq(xyzp^{2})^{-3/10},$ ⓘ Symbols: $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$: symmetric elliptic integral of third kind Permalink: http://dlmf.nist.gov/19.24.E11 Encodings: TeX, pMML, png See also: Annotations for §19.24(ii), §19.24(ii), §19.24 and Ch.19
 19.24.12 $\tfrac{1}{3}(\sqrt{x}+\sqrt{y}+\sqrt{z})\leq R_{G}\left(x,y,z\right)\leq\min% \left(\sqrt{\frac{x+y+z}{3}},\frac{x^{2}+y^{2}+z^{2}}{3\sqrt{xyz}}\right).$ ⓘ Symbols: $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of second kind Permalink: http://dlmf.nist.gov/19.24.E12 Encodings: TeX, pMML, png See also: Annotations for §19.24(ii), §19.24(ii), §19.24 and Ch.19

Inequalities for $R_{C}\left(x,y\right)$ and $R_{D}\left(x,y,z\right)$ are included as special cases (see (19.16.6) and (19.16.5)).

Other inequalities for $R_{F}\left(x,y,z\right)$ are given in Carlson (1970).

If $a$ ($\neq 0$) is real, all components of $\mathbf{b}$ and $\mathbf{z}$ are positive, and the components of $z$ are not all equal, then

 19.24.13 $\displaystyle R_{a}\left(\mathbf{b};\mathbf{z}\right)R_{-a}\left(\mathbf{b};% \mathbf{z}\right)$ $\displaystyle>1,$ $\displaystyle R_{a}\left(\mathbf{b};\mathbf{z}\right)+R_{-a}\left(\mathbf{b};% \mathbf{z}\right)$ $\displaystyle>2;$ ⓘ Symbols: $R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_% {n}}\right)$ or $R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)$: multivariate hypergeometric function Referenced by: §19.24(i) Permalink: http://dlmf.nist.gov/19.24.E13 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §19.24(ii), §19.24(ii), §19.24 and Ch.19

see Neuman (2003, (2.13)). Special cases with $a=\pm\frac{1}{2}$ are (19.24.8) (because of (19.16.20), (19.16.23)), and

 19.24.14 $\displaystyle R_{F}\left(x,y,z\right)R_{G}\left(x,y,z\right)$ $\displaystyle>1,$ $\displaystyle R_{F}\left(x,y,z\right)+R_{G}\left(x,y,z\right)$ $\displaystyle>2.$ ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of first kind and $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of second kind Permalink: http://dlmf.nist.gov/19.24.E14 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §19.24(ii), §19.24(ii), §19.24 and Ch.19

The same reference also gives upper and lower bounds for symmetric integrals in terms of their elementary degenerate cases. These bounds include a sharper but more complicated lower bound than that supplied in the next result:

 19.24.15 $R_{C}\left(x,\tfrac{1}{2}(y+z)\right)\leq R_{F}\left(x,y,z\right)\leq R_{C}% \left(x,\sqrt{yz}\right),$ $x\geq 0$,

with equality iff $y=z$.